The 10 O'Clock Live team: Charlie Brooker, Jimmy Carr, David Mitchell and Lauren Laverne. Photograph: Linda Nylind for the Guardian

British TV producers are curiously obsessed with reproducing two formats which, for reasons of age or geography, most of the UK audience has never seen: the government-shaking BBC satire That Was The Week That Was, which ran from 1962-63, and Jon Stewart's cult American franchise The Daily Show. Channel 4's 10 O'Clock Live became the latest show to try to raise the ghost of TW3 and channel Stewart.

In a brave scheduling decision, the new show – an hour of topical material from David Mitchell, Jimmy Carr, Lauren Laverne and Charlie Brooker – will fight for the political viewing constituency with the BBC's Newsnight and Question Time, the latter responding to its rival's debut with the populist move of fielding its first current footballer pundit, Clarke Carlisle.

Although Mitchell received most of the pre-publicity, Laverne opened the show, introducing the three blokes. Apart from one spot in which the quartet round-tables the day's news, the format rotates monologues from the men, starting with a monologue from Carr in the style of the stand-up intro traditional for America's late-night TV hosts.

Carr had been tweeting about his high state of nerves, but he is a very experienced live performer and his riff was impressively slick. Whereas many new magazine series begin with material clearly rehearsed and tested during pilots – and Brooker's later "Who is Sarah Palin?" could have been in the works for a year – Carr started with gags on Alan Johnson's resignation that could only have been written this evening.

These centred on genital double-entendres – "Johnson out, Balls in!" – but the monologue continued with genuinely savage sections on al-Qaida and paedophile priests, although Carr's sign-off phrase – "And that was the news this week" – sounded uneasily phrased to avoid plagiarising directly the TW3 title.

We had been warned to "expect very strong language" and the first f-word came, from Brooker, at 10.18pm, with Mitchell trumping it with a deliberate "Jeremy Hunt" mispronunciation at 10.31pm.

Channel 4's late-night schedules have historically been aimed at people coming in from the pub and the studio audience often sounded as if they had made the same journey, whooping at the rudest gags and booing guest David Willetts.

Mitchell, who twice referred to the atmosphere resembling a "panto", was impressively combative in his interrogation of the higher education minister, on the subject of tuition fees, and it will be interesting to see him with a politician less emollient in his responses than Willetts.

Several production adjustments would, though, be advisable before next week. While the publicity has stressed liveness and topicality, too many of the topics could have been locked-off by Monday morning and Laverne gets much less to do than the boys, which feels retrograde. And the over-dominant tone, especially from Mitchell and Brooker, is exaggerated comic rant: Basil Fawlty as reporter. The four regulars also all seem united in a liberal agenda.

But this was a highly promising, confident debut which contains sketches – such as Carr's spoof tourist guide to post-revolutionary Tunisia – that it is hard to imagine being sanctioned at the BBC, since the Jonathan Ross scandal. Ten O'Clock Live isn't yet The Daily Show but has the potential to become a must-see weekly show looking back at the week that was.